# Questions tagged [copula]

A copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the unit interval.

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$\newcommand{\Exp}[1]{\mathbb{E}\left[#1\right]}$ I am interested in understanding wether the following approach holds when calculating the expectation of two correlated random variables. Suppose $X\... • 1,002 0 votes 0 answers 12 views ### Kendall 's tau in high dimensional setting Often I see that the Kendall's tau is defined for 2 random variables by using their bivariate copula$C(u_1, u_2)$I wonder in the case of multidimensional setting (i.e. from 3 dimensions), how the ... 4 votes 1 answer 59 views ### Proving that copulas are Lipschitz continuous A copula is a function$C:[0,1]^2\to[0,1]$such that$C(x,0)=C(0,x)=0$for all$x\in[0,1]$,$C(x,1)=C(1,x)=x$for all$x\in[0,1]$, and \label{ineq} C(x_2,y_2)-C(x_1,y_2)-C(x_2,y_1)+C(... • 16.2k 0 votes 1 answer 19 views ### Calculating the joint cumulative distribution function from a junction tree Assume I have the following Junction Tree between random variables$X_1,\dots,X_7$that exactly describes the sets variables with non-zero Mutual Information (Alternatively it's the last tree in a ... • 2,118 5 votes 0 answers 166 views ### Product of correlated random variables and its transformation There is an interesting result, saying that if$Z_1, Z_2$are standard normal random variables with a correlation$\rho\in (-1,1)$, then the product$Z=Z_1Z_2$has a density function explicitly given ... 0 votes 0 answers 19 views ### What is the origin of the name "elliptical" copula? Could you please explain me why the eliptical copula is called "eliptical" ? I have read many articles but the authors assumed the knowledge of where the elliptical copula comes from is ... 1 vote 0 answers 33 views ### Combining conditional probabilities using an unconditional copula First: just a bit of background on copulas. Suppose we have a pair of continuous random variables$Y_1, Y_2$with distribution functions$F_1(y_1)=P(Y_1\leq y_1)$and$F_2(y_2)=P(Y_2\leq y_2)$. Let ... 0 votes 0 answers 25 views ### Kendall's Tau of FGM Copula The Fairly-Gumbel-Morgenstern Copula is defined as$C_\theta(u,v) = uv + \theta uv(1-u)(1-v)$With (u,v)$\in[0,1]^2$and$\theta \in [-1,1]$in order to$C_\theta$to be a copula. I know that from ... 2 votes 1 answer 58 views ### Proving that$C(x_1,x_2)=\max\{x_1+x_2-1,0\}$is a copula I have the following problem. I need to prove the copula property for the function$C(x_1,x_2)=\max\{x_1+x_2-1,0\}$, better known as the lower Fréchet-Hoeffding-Boundary. A copula is defined as a ... • 85 1 vote 0 answers 15 views ### Frequently used methods to estimate copula of discrete random variable in high dimension I am new to the estimation of copula with discrete marginal distributions. For example, I have the data of 30 discrete random count variables from$X_1$to$X_{30}$, some$X_j$has Poisson ... 0 votes 0 answers 29 views ### Question on the Farlie–Gumbel–Morgenstern copula To compute the joint bivariate probability density function (PDF) given marginal PDFs we can use the copula. The Sklar's theorem imply that copulas can be used to express a multivariate distribution ... • 1,917 0 votes 0 answers 31 views ### Uniqueness of copula for discrete random variable explanation The Sklar's theorem of copula states that Let$H$be a two-dimensional distribution function with marginal distribution functions$F$and$G$. Then there exists a copula$C$such that$H(x,y)=C(F(x),...
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Most copula families (e.g. Archimedean copulae) seem to only parameterize positive quadrant dependent (PQD) copulae. Suppose we have a PQD copula $C(u, v)$. Can we obtain a negative quadrant dependent ...
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### Marshall -olkin Copula and absolutely continuous

the Marshall -olkin copula is given by $$C(x,y)= \min(x^{1/3}y , xy^{3/7}).$$ I wonder if this copula has a density function, because I had tried find density of MO copula in R: ...
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### Kendall’s tau between $X$ and $1/Y$

I'm currently studying for my statistics exam by doing exercises from the book Statistics and Data Analysis for Financial Engineering with R examples. I'm struggling with this exercise from chapter 8: ...
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### Proof that Pearson correlation is/isn't a functional of the copula for a given pair of random variables

I have a growing interest and respect for the subject of copulas initially thanks to comments made on stats.SE by kjetil-b-halvorsen. The most interesting to me right now is the following: "[T]...
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### Sampling from Gaussian Copula with a conditional distribution approach

In the paper "Cheng and al. (2007)" on pages 193-194, the authors propose an algorithm that generates variables with a given copula function $C$ being the joint distribution. This procedure ...
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### Life expantancy of correlated light bulb batches

I have 3 light bulbs from different build batches, they have a rate of failure of 0.02, 0.05 and 0.09 each month. The failure rate between the batches is correlated, the failures are normally ...
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### Showing that the lower bivariate Fréchet-Hoeffding bound is a copula.

I want to show that the bivariate Fréchet-Hoefdding lower bound is indeed a copula. The bivariate function is defined by: $W(x,y)=\max\{x+y-1,0\}, \ x,y\in[0,1]$ Definition "Copula": A two-...
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### How to deal with copulas?

I want to model a couple of variables $(X,Y)$ using copulas. My idea is to model the marginal distributions and then use copulas to combine marginal distributions and copula to get a joint ...
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### Kendall's tau for archimedian copula

Consider a 2-dimensional archimedian copula with generator $\psi$ and $\gamma:=\psi^{-1}$, i.e. $$C(u,v)=\gamma(\psi(u)+\psi(v))$$ I want to show that $$\tau=1-\int_{0}^{\infty}t\gamma'(t)^2dt$$ I ...
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From two Independent Normal Random Variables: $X \sim N(\mu_1,\sigma)$ and $Y\sim N(\mu_1,\sigma)$, I created two DEPENDENT random variables $Z$ and $W$: $Z= X - Y$ $W= X - g(Y)$ where $g(\... • 1 0 votes 0 answers 56 views ### How to compare the entropy of copulas? Three different bivariate copulas are shown below with increasing degrees of dependence (parameter$\theta$). Differential entropy is a measure of disorder in a probability density like the copula. ... • 1,354 1 vote 1 answer 95 views ### Impact of copula choice on quantiles (sum of random variables) I am trying to get my head around the impact of different dependence structures (copulas) on the risk (quantiles) of a sum of dependent random variables (with arbitrary marginals). In a multivariate ... • 13 0 votes 1 answer 114 views ### Why is copula entropy negative? Most statistical measures are non-negative. Shannon entropy,$H(X)$, a statistical measure of disorder or uncertainty in probability distributions, is also non-negative, despite it having a negative ... • 1,354 2 votes 1 answer 116 views ### Why do elliptical copula densities contain$x_1$and$x_2$, but Archimedean copula densities contain$u_1$and$u_2$? $$c\left(u_{1}, u_{2}\right)=\frac{1}{\sqrt{1-\rho_{12}^{2}}} \exp \left\{-\frac{\rho_{12}^{2}\left(x_{1}^{2}+x_{2}^{2}\right)-2 \rho_{12} x_{1} x_{2}}{2\left(1-\rho_{12}^{2}\right)}\right\}$$ is the ... • 1,354 2 votes 1 answer 146 views ### How to simplify out of the gamma function$\Gamma(z)$? $$\mathrm{d}t(x_i; \nu) = \frac{\Gamma \bigg(\frac{\nu+1}{2}\bigg) }{\Gamma (\frac{\nu}{2}) \sqrt{\pi\nu}} \Bigg( 1+\frac{x_i^2}{\nu} \Bigg)^{-\frac{\nu+1}{2} } \enspace, i=1,2$$ How to derive a ... • 1,354 2 votes 1 answer 652 views ### Derivation of bivariate Gaussian copula density The multivariate Gaussian copula density, derived here, is $$c(u_1,\ldots,u_n;\Sigma)=|\Sigma|^{-\frac{1}{2}}\exp\!\left(-\frac{1}{2}x^{\top}(\Sigma^{-1}-I)x\right)$$ where$\Sigma$is the covariance ... • 1,354 0 votes 1 answer 83 views ### Integrate$\int_{0}^{1} u^{\frac{\alpha}{\beta}-\alpha}(1-\alpha) \log \left((1-\alpha) u^{-\alpha}\right) d u $Where$c(u, v)=u^{-\alpha}(1-\alpha) is the Marshall-Olkin copula density, we have the following integral: \begin{align} I_{1} &=\iint_{E} c(u, v) \log c(u, v) d u d v \\ &=\int_{0}^{1} \... • 1,354 1 vote 0 answers 89 views ### Entropy of the bivariate Gaussian copula: Closed-form analytical solution Background The bivariate Gaussian copula function is $$C_{\rho}(u,v)=∫_{-∞}^{Φ^{-1}(u)}∫_{-∞}^{Φ^{-1}(v)}\frac{1}{2π\sqrt{1-ρ^2}}×exp⁡(-\frac{x^2+y^2-2ρxy}{2(1-ρ^2)})dxdy.$$ The multivariate Gaussian ... • 1,354 3 votes 1 answer 64 views ### How to simplify function to\log (1-\theta)+\frac{\theta}{2-\theta}? \begin{align} f(\theta)& = \log \frac{1}{1-\theta}-\frac{\theta}{2-\theta}+\frac{\theta^{2}}{(2-\theta)^{2}} \\ & = \log (1-\theta)+\frac{\theta}{2-\theta} \end{align} How are the two ... • 1,354 1 vote 1 answer 41 views ### Do copulas preserve uniform convergence? Suppose I have a distributionQ$of$\left(Y_i, X_i\right)$, such that$Q(y,x)=C(F_Y(y), F_X(x))$, for some copula$C$and marginal CDFs$F_X$and$F_Y$, by Sklar's theorem. Now, suppose I have ... • 1,482 2 votes 1 answer 58 views ### Showing that a function is a (family of) copula(s) I am currently working with the book An Introduction to Copulas by Roger Nelsen. I am having trouble solving one of the exercises in the first chapter. The exercise reads as follows: Let$C$be a ... • 305 3 votes 0 answers 173 views ### Simplify$\mathbb{E}\left[ -\log c(u,v)\right] $, the expected logarithm of the copula density 2020 11.26: I finished the derivation but haven't posted it here. I leave this open so that those willing to try their own version can if they want Question How can we derive a closed-form analytical ... • 1,354 2 votes 1 answer 168 views ### Archimedean Clayton copula entropy Question I would like to derive the entropy of Archimedean parametric copulas (Clayton, Frank, or Gumbel), focusing here on the Clayton copula. Link to similar question on the t-copula. The bivariate ... • 1,354 1 vote 0 answers 23 views ### Gluing copulas along a curve Let$C(u,v)$be a 2-copula and$W(u,v)=\max(u+v-1,0)$. I want to glue$C$with$W$along the curve$u^2+v^2=1$s.t $$D(u,v)=\begin{cases}u+v-1,& u^2+v^2\ge 1,\\ C(u,v), &u^2+v^2<1 \end{... • 53 1 vote 0 answers 38 views ### Find a joint distribution function with known Kendall's tau (Copula) Assume we have two random variables X_1\sim Exp(2), X_2\sim Exp(1/2). I need to find a joint distribution function, so that Kendall's Tau will be \rho_\tau(X_1,X_2)=-0.85. I know that somehow I ... 0 votes 1 answer 63 views ### What is the correlation of a copula of a multivariate normal distribution? [duplicate] If X_1, X_2 \sim \mathcal{N}(0,1) with correlation \rho, and Y_1 = F_{X_1}(X_1) and Y_2 = F_{X_2}(X_2), what is the correlation between Y_1 and Y_2? What is a proof that the correlation of ... 2 votes 1 answer 75 views ### Copula with a certain correlation What does it mean for values to be "drawn from a normal copula with correlation \rho\in [0, 1]"? Is that a normal distribution with a covariance matrix whose entries are uniform random in ... 1 vote 0 answers 48 views ### Archimedean Copula Probabilities I have the following excercise : Let C be an Archimedean copula with generator given by \psi(x) = E[e^{ −xV} ], where V is an exponentially distributed random variable with expectation 1. Calculate ... • 55 1 vote 1 answer 64 views ### Copula Theory : CDF from Marginals I have given (X,Y) to be a two-dimensional random vector with Exp(1)-marginals and a Copula C(u,v) = uv + (1-u)(1-v)uv I need to determine the density of (X,Y), if it exists. I would assume that ... • 55 1 vote 0 answers 65 views ### Calculating Kendall's Tau (Copulas) Let (X_1,X_2) be a random vector with X_1 \sim Exp(1) and X_2 \sim N(0,1). The dependence structure is given by the copula$$C(u_1,u_2)=\frac{1}{3}W(u_1,u_2)+\frac{2}{3}\Pi(u_1,u_2), \ \ \ u \in ... 0 votes 1 answer 43 views ### Joint probability from marginal and relation between variables I would like to know whether it is possible to obtain the joint density function$p(x,y)$if I know one marginal, which is Gaussian$p(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp(-\frac{1}{2} (\frac{x - \mu}...
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I am working on a data set with the Gaussian and the Student-t copulas and I need to define their derivatives. For the Gaussian copula I have defined it as follows:  \frac{d}{du}C^{Gauss}_{\alpha}= \...
Consider two uniform random variables $U, V$ . If $U \sim \mbox{Uniform}(0,1)$ and $V = 1 -U$ how would we show that the joint distribution of $(U, V)$ is given by $\max \{ u + v -1,0 \}$? The ...